Finite Group Databasebeta

← #27279680

Subgroups — #27279680

Order 13568.

674 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810680C211yesmaximal, characteristic, complemented
2#20283046C211yesmaximal, characteristic
3C2×#16810356C212yesmaximal
4#20272594C212yesmaximal, complemented
5C53×#1434C211yesmaximal, characteristic, complemented
6C2×#13133200—21no—
7C2×#13133191C2211yescharacteristic
8#16810680—24no—
9C2×C4⋊#88322—21no—
10C2×C22⋊#88322C2212yes—
11C2×#13133128—21no—
12#16810525C411yescharacteristic, complemented
13#16810525C411yescharacteristic, complemented
14#16810521C411yescharacteristic, complemented
15#16810521C411yescharacteristic, complemented
16#16810514—21no—
17#16810514—21no—
18#16810502—21no—
19#16810483C2211yescharacteristic
20#16810356—24no—
21C22×C42×C53C2211yescharacteristic
22C2×#95×C53—21no—
23C2×#95×C53—21no—
24C2×#95×C53C2212yes—
25C22×D212C2×C411yescharacteristic
26C22×C22⋊D53—21no—
27C22×#88322—22no—
28#13133200—44no—
29#13133191—22no—
30#13133191—22no—
31C2×C4⋊Dic53C2×C411yescharacteristic
32C2×C4⋊Dic53C2×C411yescharacteristic
33C2×C4⋊Dic53C2×C411yescharacteristic
34C2×#11719919—21no—
35C2×#11719919—21no—
36C2×C22⋊Dic53—21no—
37C2×#11719921—21no—
38C2×#11719921—21no—
39C2×#11719921—21no—
40C2×#11719921—21no—
41C2×#11719921—21no—
42C4⋊#88322—24no—
43C22⋊#88322—22no—
44C22⋊#88322D414yes—
45#13133128—24no—
46#13133124—21no—
47#13133124—21no—
48#13133124—21no—
49C23×C4×C53—22no—
50C23×C4×C53D411yescharacteristic
51C23×C4×C53D411yescharacteristic
52C23×C4×C53C2311yescharacteristic
53C2×C42×C53—22no—
54C2×C42×C53—22no—
55C2×C42×C53C2×C411yescharacteristic
56C2×C42×C53C2×C411yescharacteristic
57C2×C42×C53D411yescharacteristic
58C2×C42×C53D411yescharacteristic
59#95×C53—24no—
60#95×C53—22no—
61#95×C53—22no—
62#95×C53—42no—
63C24×C53C2×D411yescharacteristic
64C23×D53—21no—
65C22×C4×C53—22no—
66C22×C4×C53—22no—
67C22×C4×C53—21no—
68C22×C4×C53—22no—
69C22×C4×C53—21no—
70C22×C4×C53—22no—
71C22×C4×C53—24no—
72C22×C4×C53—41no—
73C22×C4×C53—41no—
74C22×C4×C53—41no—
75C22×C4×C53—42no—
76C22×C4×C53—42no—
77C22×C4×C53—42no—
78C22×C4×C53—42no—
79C22×C4×C53—42no—
80C22×C4×C53—42no—
81C22×C4×C53C22⋊C411yescharacteristic
82C22×C4×C53C22⋊C411yescharacteristic
83C22×C4×C53C22⋊C411yescharacteristic
84C22×C4×C53C22⋊C411yescharacteristic
85C22×C4×C53C22×C411yescharacteristic
86C22×C4×C53C2×D411yescharacteristic
87C22×C4×C53C2×D412yes—
88C22×C4×C53#4112yes—
89C2×C22⋊D53—42no—
90C2×C22⋊D53—42no—
91C2×C22⋊D53—41no—
92C2×C22⋊D53—41no—
93C22×Dic53—21no—
94C22×Dic53—21no—
95C22×Dic53—21no—
96C2×D212—22no—
97C2×D212—41no—
98C2×D212—41no—
99C42×C53—22no—
100C42×C53—22no—
101C42×C53—21no—
102C42×C53—21no—
103C42×C53—21no—
104C42×C53—21no—
105C22⋊Dic53—42no—
106#11719921—42no—
107#11719921—42no—
108#11719921—42no—
109#11719921—42no—
110#11719921—41no—
111#11719921—41no—
112C4⋊Dic53—22no—
113C4⋊Dic53—41no—
114C4⋊Dic53—41no—
115#11719919—42no—
116#11719919—42no—
117C2×#88322—24no—
118C2×#88322—42no—
119C2×#88322—42no—
120C23×C53—21no—
121C23×C53—21no—
122C23×C53—21no—
123C23×C53—21no—
124C23×C53C2×C22⋊C411yescharacteristic
125C23×C53#12112yes—
126C23×C53AGL(1,ℤ/8)12yes—
127C23×C53AGL(1,ℤ/8)11yescharacteristic
128C23×C53#13711yescharacteristic
129C22×D53—41no—
130C22×D53—41no—
131C22×D53—41no—
132C22×D53—41no—
133C22×D53—41no—
134C22×D53—41no—
135C22×D53—41no—
136C2×C4×C53—21no—
137C2×C4×C53—21no—
138C2×C4×C53—21no—
139C2×C4×C53—21no—
140C2×C4×C53—21no—
141C2×C4×C53—21no—
142C2×C4×C53—21no—
143C2×C4×C53—21no—
144C2×C4×C53—21no—
145C2×C4×C53—21no—
146C2×C4×C53—21no—
147C2×C4×C53—21no—
148C2×C4×C53—21no—
149C2×C4×C53—21no—
150C2×C4×C53—21no—
151C2×C4×C53—42no—
152C2×C4×C53—42no—
153C2×C4×C53—44no—
154C2×C4×C53—42no—
155C2×C4×C53—42no—
156C2×C4×C53—44no—
157C2×C4×C53—44no—
158C2×C4×C53—44no—
159C2×C4×C53—82no—
160C2×C4×C53—82no—
161C2×C4×C53—82no—
162C2×C4×C53—82no—
163C2×C4×C53#10412yes—
164C2×C4×C53C4×D412yes—
165C2×C4×C53#12011yescharacteristic
166C2×C4×C53#12311yescharacteristic
167C22⋊D53—82no—
168C22⋊D53—82no—
169C2×Dic53—41no—
170C2×Dic53—41no—
171C2×Dic53—41no—
172C2×Dic53—41no—
173C2×Dic53—41no—
174C2×Dic53—41no—
175C2×Dic53—41no—
176C2×Dic53—41no—
177C2×Dic53—41no—
178D212—82no—
179#88322—44no—
180#9118—531nomaximal, Sylow
181C22×C53—22no—
182C22×C53—22no—
183C22×C53—22no—
184C22×C53—21no—
185C22×C53—21no—
186C22×C53—21no—
187C22×C53—21no—
188C22×C53—22no—
189C22×C53#38611yescharacteristic
190C22×C53#41011yescharacteristic
191C22×C53#41011yescharacteristic
192C22×C53#41711yescharacteristic
193C22×C53#41811yescharacteristic
194C22×C53C2×#10411yescharacteristic
195C22×C53#42111yescharacteristic
196C22×C53#46812yes—
197C22×C53#46912yes—
198C2×D53—81no—
199C2×D53—81no—
200C2×D53—81no—
201C2×D53—81no—
202C2×D53—81no—
203C2×D53—81no—
204C2×D53—81no—
205C4×C53—22no—
206C4×C53—21no—
207C4×C53—22no—
208C4×C53—21no—
209C4×C53—41no—
210C4×C53—41no—
211C4×C53—41no—
212C4×C53—84no—
213Dic53—81no—
214Dic53—81no—
215Dic53—81no—
216C2×#432—531no—
217#2854—531no—
218C2×#327—532no—
219#1434D5311yescharacteristic, complemented
220#1294—532no—
221C2×C53—22no—
222C2×C53—22no—
223C2×C53#150511yescharacteristic
224C2×C53#150911yescharacteristic
225C2×C53#151211yescharacteristic
226C2×C53#151411yescharacteristic
227C2×C53#187111yescharacteristic
228C2×C53#189511yescharacteristic
229C2×C53#189911yescharacteristic
230D53—161no—
231C2×#123—1061no—
232C2×#121—531no—
233C22×C42C2×D5311yescharacteristic
234#432—1064no—
235C2×C4⋊C8—1061no—
236C2×C22⋊C8—532no—
237C2×#97—1061no—
238#390—531no—
239#390—531no—
240#389—531no—
241#389—531no—
242#388—1061no—
243#388—1061no—
244#386—1061no—
245#380—531no—
246C2×#95—21no—
247C2×#95—21no—
248C2×#95C2×D5312yes—
249#327—1064no—
250C53#911811yescharacteristic, complemented, Sylow
251C22×D4—531no—
252C22×D4—1061no—
253C23×C4—22no—
254C23×C4C22⋊D5311yescharacteristic
255C23×C4C22⋊D5311yescharacteristic
256C23×C4C22×D5311yescharacteristic
257C22×C8—1062no—
258#123—2124no—
259#121—1062no—
260#121—1062no—
261C2×C4⋊C4—531no—
262C2×C4⋊C4—531no—
263C2×C4⋊C4—531no—
264C2×C4⋊C4—1061no—
265C2×C4⋊C4—1061no—
266C2×C22⋊C4—1061no—
267C2×C22⋊C4—1061no—
268C2×C22⋊C4—1061no—
269C2×C22⋊C4—1061no—
270C2×C22⋊C4—1061no—
271C2×C22⋊C4—1061no—
272C2×C42—22no—
273C2×C42—22no—
274C2×C42C4×D5311yescharacteristic
275C2×C42C4×D5311yescharacteristic
276C2×C42D21211yescharacteristic
277C2×C42D21211yescharacteristic
278C4⋊C8—1064no—
279C22⋊C8—534no—
280C22⋊C8—1062no—
281#97—1064no—
282#95—24no—
283#95—22no—
284#95—22no—
285#95—42no—
286#95—1061no—
287#95—1061no—
288#95—1061no—
289C24—1061no—
290C24C2×C22⋊D5311yescharacteristic
291C2×D4—1062no—
292C2×D4—2121no—
293C2×D4—2121no—
294C2×D4—2122no—
295C2×D4—2122no—
296C2×D4—2121no—
297C2×D4—2121no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—22no—
304C22×C4—24no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—1061no—
315C22×C4—1061no—
316C22×C4—1061no—
317C22×C4#1171992111yescharacteristic
318C22×C4#1171992111yescharacteristic
319C22×C4#1171992111yescharacteristic
320C22×C4#1171992111yescharacteristic
321C22×C4C2×C4×D5311yescharacteristic
322C22×C4C2×D21211yescharacteristic
323C22×C4D4×D5312yes—
324C22×C4#1171994712yes—
325C2×C8—1064no—
326C2×C8—2122no—
327C2×C8—2122no—
328C4⋊C4—1062no—
329C4⋊C4—2121no—
330C4⋊C4—2121no—
331C4⋊C4—2122no—
332C4⋊C4—2122no—
333C22⋊C4—2122no—
334C22⋊C4—2122no—
335C22⋊C4—2122no—
336C22⋊C4—2122no—
337C22⋊C4—2122no—
338C22⋊C4—2121no—
339C22⋊C4—2121no—
340C42—22no—
341C42—22no—
342C42—21no—
343C42—21no—
344C42—21no—
345C42—21no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23—21no—
350C23—2121no—
351C23—2121no—
352C23—2121no—
353C23—2121no—
354C23—2121no—
355C23—2121no—
356C23—2121no—
357C23C2×#1171992111yescharacteristic
358C23#1313318912yes—
359C23#1313324212yes—
360C23#1313324311yescharacteristic
361C23#1313324811yescharacteristic
362D4—4242no—
363D4—4242no—
364D4—4242no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—44no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—2121no—
393C2×C4—2121no—
394C2×C4—2121no—
395C2×C4—2121no—
396C2×C4—2121no—
397C2×C4—2121no—
398C2×C4—2121no—
399C2×C4—2121no—
400C2×C4—2121no—
401C2×C4#1313314512yes—
402C2×C4#1313317712yes—
403C2×C4#1313318711yescharacteristic
404C2×C4#1313320111yescharacteristic
405C8—2124no—
406C22—22no—
407C22—22no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—22no—
413C22—22no—
414C22—4241no—
415C22—4241no—
416C22—4241no—
417C22—4241no—
418C22—4241no—
419C22—4241no—
420C22—4241no—
421C22#1681050011yescharacteristic
422C22#1681059611yescharacteristic
423C22#1681059611yescharacteristic
424C22#1681062211yescharacteristic
425C22#1681062711yescharacteristic
426C22C2×#1313314511yescharacteristic
427C22#1681064111yescharacteristic
428C22#1681087112yes—
429C22#1681087412yes—
430C4—22no—
431C4—21no—
432C4—22no—
433C4—21no—
434C4—41no—
435C4—41no—
436C4—41no—
437C4—84no—
438C4—4241no—
439C4—4241no—
440C4—4241no—
441C2—22no—
442C2—22no—
443C2—8481no—
444C2#2027322211yescharacteristic
445C2#2027323811yescharacteristic
446C2#2027325111yescharacteristic
447C2#2027326111yescharacteristic
448C2#2027501511yescharacteristic
449C2#2027515111yescharacteristic
450C2#2027517111yescharacteristic
subgroupG/H
centerC23C2×#11719921
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#1434C2
socleC23×C53C2×C22⋊C4
prime radicalC2×#13133191C22
cosocleC23×C4×C53C23
hypercenter#1434D53
hypocenterC53#9118
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1434

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#16810680C2semidirect
4. #20272594C2semidirect
5. C53×#1434C2semidirect
12. #16810525C4semidirect
13. #16810525C4semidirect
14. #16810521C4semidirect
15. #16810521C4semidirect
219. #1434D53semidirect
250. C53#9118semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#168106802
#202830462
C2×#168103562
#202725942
C53×#14342
#911853